Multi-model Estimation with J-linkage

1. Multi-model Estimation with J-linkage Jeongkyun Lee

2. Motivation How do we find parameters of a model that contains outliers? • Application in vision: geometric figure fitting, planar surface detection, motion segmentation, etc.

3. Single-model Estimation • Least Squares • Least Median of Squares (LMedS) • Random Sample Consensus (RANSAC) • M-SAC • MLESAC • PROSAC • Etc.

4. Least Squares • Calculate parameters of model function • Overdetermined data set • Minimized sum of squared residuals with a matrix form,

5. Least Squares Without outliers With outliers

6. RANSAC • Iterative method • Non-deterministic • Robust fitting in the presence of outliers • Simple algorithm Algorithm • M. A. Fischler, R. C. Bolles. Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography. Comm. of the ACM, Vol 24, pp 381-395, 1981.

7. RANSAC

8. Multi-model Estimation • Residual histogram analysis (RHA) • Sequential RANSAC • Multi-RANSAC • J-linkage • Kernel fitting (KF) • Mean-shift (MS) • Etc.

9. Multi-model Estimation with J-Linkage • Fit multiple structures simultaneously • Require no initial parameters: # of models, model parameters Algorithm Given N points, Generate M model hypothesis (Random sampling) Build a N x M matrix, comprised of Preference Sets of points J-linkage clustering

10. Multi-model Estimation with J-Linkage Preference Set

11. Multi-model Estimation with J-Linkage • Random Sampling • A minimal sample set (MSS) is constructed in a way that neighbouring points are selected with higher probability. • One sample is selected with uniform probability • If a point is given, then has the following probability: Z is a normalized constant, is chosen heuristically.

12. Multi-model Estimation with J-Linkage • J-linkage Clustering • Starting from all singletons • Each sweep of the algorithm merges the two clusters with the smallest distance Measure the degree of overlap of the two sets and ranges from 0 (identical sets) to 1 (disjoint sets)

13. Multi-model Estimation with J-Linkage • J-linkage Clustering Assumption One-to-one matching between a point and a model Algorithm

14. Multi-model Estimation with J-Linkage • Example 1 2 3 4

15. Multi-model Estimation with J-Linkage • Results

16. Multi-model Estimation with J-Linkage • Results

17. Multi-model Estimation with J-Linkage • Results

18. Multi-model Estimation with J-Linkage • Results

19. Multi-model Estimation with J-Linkage • Other Results 1 David F. Fouhey, “Multi-model Estimation in the Presence of Outliers”

20. Multi-model Estimation with J-Linkage • Other Results 1 David F. Fouhey, “Multi-model Estimation in the Presence of Outliers”

21. Multi-model Estimation with J-Linkage • Other Results 2 HanziWang, “Robust Multi-Structure Fitting”, A tutorial in ACCV 2012.

22. Multi-model Estimation with J-Linkage • Other Results 2 HanziWang, “Robust Multi-Structure Fitting”, A tutorial in ACCV 2012.

23. Reference • David F. Fouhey, “Multi-model Estimation in the Presence of Outliers” • Stefano Branco, “RANSAC/MLESAC, Estimating parameters of models with outliers” • Hanzi Wang, “Robust Multi-Structure Fitting”, A tutorial in ACCV 2012.

24. Thank you!

25. Appendix

26. Appendix